On The Cauchy Problem for Non Effectively Hyperbolic Operators, The Ivrii-Petkov-Hörmander Condition and the Gevrey Well Posedness

2000 Mathematics Subject Classification: 35L15, Secondary 35L30.

Inverse Problems Involving Generalized Axial-Symmetric Helmholtz Equation

MSC 2010: 35J05, 33C10, 45D05

Bayesian and Frequentist Two-Sample Predictions of the Inverse Weibull Model Based on Generalized Order Statistics

2000 Mathematics Subject Classification: 62E16,62F15, 62H12, 62M20.

Leak identification in saturated unsteady flow via a Cauchy problem

This work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we therefore modify and employ an iterative regularizing method proposed in [1] and [2]. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results indicate that the leak zones can be identified with the proposed method.

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.

A method of fundamental solutions for theaxisymmetric backward heat equation

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

The existence of an inverse limit of inverse system of measure spaces - a purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the birth of the modern probability theory. Results from Kolmogorov [10], Bochner [2], Choksi [5], Metivier [14], Bourbaki [3] among others have paved the way of the deep understanding of the problem under consideration. All the above results, however, call for some topological concepts, or at least ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sucient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given.

Foraminiferal faunal estimates of paleotemperature: Circumventing the no-analog problem yields cool ice age tropics

The sensitivity of the tropics to climate change, particularly the amplitude of glacial-to-interglacial changes in sea surface temperature (SST), is one of the great controversies in paleoclimatology. Here we reassess faunal estimates of ice age SSTs, focusing on the problem of no-analog planktonic foraminiferal assemblages in the equatorial oceans that confounds both classical transfer function and modern analog methods. A new calibration strategy developed here, which uses past variability of species to define robust faunal assemblages, solves the no-analog problem and reveals ice age cooling of 5° to 6°C in the equatorial current systems of the Atlantic and eastern Pacific Oceans. Classical transfer functions underestimated temperature changes in some areas of the tropical oceans because core-top assemblages misrepresented the ice age faunal assemblages. Our finding is consistent with some geochemical estimates and model predictions of greater ice age cooling in the tropics than was inferred by Climate: Long-Range Investigation, Mapping, and Prediction (CLIMAP) [1981] and thus may help to resolve a long-standing controversy. Our new foraminiferal transfer function suggests that such cooling was limited to the equatorial current systems, however, and supports CLIMAP's inference of stability of the subtropical gyre centers.

Towards NIR emission: 2-(1H-tetrazol-1-yl)pyridine as versatile ligand for the synthesis of two new families of cationic and neutral Ir(III) complexes.

In the last decades, cyclometalated Ir(III) complexes have drawn a large interest for their unique properties: they are excellent triplet state emitters, thus the emission is phosphorescent in nature; typically high quantum yields and good stability make them good candidates for luminescent materials. Moreover, through an opportune choice of the ligands, it is possible to tune the emission along the whole visible spectra. Thanks to these interesting features, Ir(III) complexes have found different applications in several areas of applied science, from OLEDs to bioimaging. In particular, regarding the second application, a remarkable red-shift in the emission is required, in order to minimize the problem of the tissue penetration and the possible damages for the organisms. With the aim of synthesizing a new family of NIR emitting Ir(III) complexes, we envisaged the possibility to use for the first time 2-(1H-tetrazol-1-yl)pyridine as bidentate ligand able to provide the required red-shift of the emission of the final complexes. Exploiting the versatility of the ligand, I prepared two different families of heteroleptic Ir(III) complexes. In detail, in the first case the 2-(1H-tetrazol-1-yl)pyridine was used as bis-chelating N^N ligand, leading to cationic complexes, while in the second case it was used as cyclometalating C^N ligand, giving neutral complexes. The structures of the prepared molecules have been characterised by NMR spectroscopy and mass spectrometry. Moreover, the neutral complexes’ emissive properties have been measured: emission spectra have been recorded in solution at both room temperature and 77K, as well as in PMMA matrix. DFT calculation has then been performed and the obtained results have been compared to experimental ones.

Tensor Completion for Multidimensional Inverse Problems with Applications to Magnetic Resonance Relaxometry

This thesis deals with tensor completion for the solution of multidimensional inverse problems. We study the problem of reconstructing an approximately low rank tensor from a small number of noisy linear measurements. New recovery guarantees, numerical algorithms, non-uniform sampling strategies, and parameter selection algorithms are developed. We derive a fixed point continuation algorithm for tensor completion and prove its convergence. A restricted isometry property (RIP) based tensor recovery guarantee is proved. Probabilistic recovery guarantees are obtained for sub-Gaussian measurement operators and for measurements obtained by non-uniform sampling from a Parseval tight frame. We show how tensor completion can be used to solve multidimensional inverse problems arising in NMR relaxometry. Algorithms are developed for regularization parameter selection, including accelerated k-fold cross-validation and generalized cross-validation. These methods are validated on experimental and simulated data. We also derive condition number estimates for nonnegative least squares problems. Tensor recovery promises to significantly accelerate N-dimensional NMR relaxometry and related experiments, enabling previously impractical experiments. Our methods could also be applied to other inverse problems arising in machine learning, image processing, signal processing, computer vision, and other fields.

Which-path problem for one and two particles with two degrees of freedom and a relation between transverse spatial structure and group velocity of light

Quantum mechanics, optics and indeed any wave theory exhibits the phenomenon of interference. In this thesis we present two problems investigating interference due to indistinguishable alternatives and a mostly unrelated investigation into the free space propagation speed of light pulses in particular spatial modes. In chapter 1 we introduce the basic properties of the electromagnetic field needed for the subsequent chapters. In chapter 2 we review the properties of interference using the beam splitter and the Mach-Zehnder interferometer. In particular we review what happens when one of the paths of the interferometer is marked in some way so that the particle having traversed it contains information as to which path it went down (to be followed up in chapter 3) and we review Hong-Ou-Mandel interference at a beam splitter (to be followed up in chapter 5). In chapter 3 we present the first of the interference problems. This consists of a nested Mach-Zehnder interferometer in which each of the free space propagation segments are weakly marked by mirrors vibrating at different frequencies [1]. The original experiment drew the conclusions that the photons followed disconnected paths. We partition the description of the light in the interferometer according to the number of paths it contains which-way information about and reinterpret the results reported in [1] in terms of the interference of paths spatially connected from source to detector. In chapter 4 we briefly review optical angular momentum, entanglement and spontaneous parametric down conversion. These concepts feed into chapter 5 in which we present the second of the interference problems namely Hong-Ou-Mandel interference with particles possessing two degrees of freedom. We analyse the problem in terms of exchange symmetry for both boson and fermion pairs and show that the particle statistics at a beam splitter can be controlled for suitably chosen states. We propose an experimental test of these ideas using orbital angular momentum entangled photons. In chapter 6 we look at the effect that the transverse spatial structure of the mode that a pulse of light is excited in has on its group velocity. We show that the resulting group velocity is slower than the speed of light in vacuum for plane waves and that this reduction in the group velocity is related to the spread in the wave vectors required to create the transverse spatial structure. We present experimental results of the measurement of this slowing down using Hong-Ou-Mandel interference.

Leptospirosis in northern Tanzania: exploring the role of rodents and ruminant livestock in a neglected public health problem

Leptospirosis is an important but neglected zoonotic disease that is often overlooked in Africa. Although comprehensive data on the incidence of human disease are lacking, robust evidence of infection has been demonstrated in people and animals from all regions of the continent. However, to date, there are few examples of direct epidemiological linkages between human disease and animal infection. In East Africa, awareness of the importance of human leptospirosis as a cause of non-malarial febrile illness is growing. In northern Tanzania, acute leptospirosis has been diagnosed in 9% of patients with severe febrile illness compared to only 2% with malaria. However, little is known about the relative importance of different potential animal hosts as sources of human infection in this area. This project was established to investigate the roles of rodents and ruminant livestock, important hosts of Leptospira in other settings, in the epidemiology of leptospirosis in northern Tanzania. A cross-sectional survey of rodents living in and around human settlements was performed alongside an abattoir survey of ruminant livestock. Unusual patterns of animal infection were detected by real-time PCR detection. Renal Leptospira infection was absent from rodents but was detected in cattle from several geographic areas. Infection was demonstrated for the first time in small ruminants sub-Saharan Africa. Two major Leptospira species and a novel Leptospira genotype were detected in livestock. L. borgpetersenii was seen only in cattle but L. kirschneri infection was detected in multiple livestock species (cattle, sheep and goats), suggesting that at least two distinct patterns of Leptospira infection occur in livestock in northern Tanzania. Analysis of samples from acute leptospirosis in febrile human patients could not detect Leptospira DNA by real-time PCR but identified social and behavioural factors that may limit the utility of acute-phase diagnostic tests in this community. Analysis of serological data revealed considerable overlap between serogroups detected in cattle and human leptospirosis cases. Human disease was most commonly attributed to the serogroups Mini and Australis, which were also predominant reactive serogroups in cattle. Collectively, the results of this study led to the hypothesis that livestock are an important reservoir of Leptospira infection for people in northern Tanzania. These results also challenge our understanding of the relationship between Leptospira and common invasive rodent species, which do not appear to maintain infection in this setting. Livestock Leptospira infection has substantial potential to affect the well-being of people in East Africa, through direct transmission of infection or through indirect effects on food production and economic security. Further research is needed to quantify the impact of livestock leptospirosis in Africa and to develop effective interventions for the control of human and animal disease.

Investigating problem-based learning in Saudi Arabian mathematics education: a TIMSS-related study

The aim of this study is to investigate the effectiveness of problem-based learning (PBL) on students’ mathematical performance. This includes mathematics achievement and students’ attitudes towards mathematics for third and eighth grade students in Saudi Arabia. Mathematics achievement includes, knowing, applying, and reasoning domains, while students’ attitudes towards mathematics covers, ‘Like learning mathematics’, ‘value mathematics’, and ‘a confidence to learn mathematics’. This study goes deeper to examine the interaction of a PBL teaching strategy, with trained face-to-face and self-directed learning teachers, on students’ performance (mathematics achievement and attitudes towards mathematics). It also examines the interaction between different ability levels of students (high and low levels) with a PBL teaching strategy (with trained face-to-face or self-directed learning teachers) on students’ performance. It draws upon findings and techniques of the TIMSS international benchmarking studies. Mixed methods are used to analyse the quasi-experimental study data. One -way ANOVA, Mixed ANOVA, and paired t-tests models are used to analyse quantitative data, while a semi-structured interview with teachers, and author’s observations are used to enrich understanding of PBL and mathematical performance. The findings show that the PBL teaching strategy significantly improves students’ knowledge application, and is better than the traditional teaching methods among third grade students. This improvement, however, occurred only with the trained face-to-face teacher’s group. Furthermore, there is robust evidence that using a PBL teaching strategy could raise significantly students’ liking of learning mathematics, and confidence to learn mathematics, more than traditional teaching methods among third grade students. Howe ver, there was no evidence that PBL could improve students’ performance (mathematics achievement and attitudes towards mathematics), more than traditional teaching methods, among eighth grade students. In 8th grade, the findings for low achieving students show significant improvement compared to high achieving students, whether PBL is applied or not. However, for 3th grade students, no significant difference in mathematical achievement between high and low achieving students was found. The results were not expected for high achieving students and this is also discussed. The implications of these findings for mathematics education in Saudi Arabia are considered.